where \(V_{-}=\frac{1}{2}(|V|-V)\)  is the negative part of \(V\)  and \(\tau \)  is the standard normalized trace of \(L_\infty (n_\theta )\) . This inequality is consistent with Lieb's version of the CLR inequality for closed manifolds [1]}, [2]}. For \(p=n/2\)  we get a CLR inequality for Schrödinger operators \(\Delta +\lambda (V)\)  with \(V=V^*\in L_{n/2}(n_\theta )\)  if \(n\ge 3\)  or with \(V=V^*\in L_q(n_\theta )\) , \(q>1\) , if \(n=2\) . The latter condition is consistent with the CLR inequality for Schrödinger operators on bounded regions of \(\mathbb {R}^2\)  by Birman-Solomyak [3]} (see also [4]}). In particular, unlike in the Euclidean space setting we do get an inequality in dimension 2.
